Deterministic Sampling on the Circle using Projected Cumulative Distributions

TL;DR

The paper addresses the challenge of efficiently representing arbitrary continuous angular densities on the circle with a deterministic Dirac mixture using an arbitrary number of samples. It extends the projected cumulatives framework to $S^1$ by projecting densities along multiple directions (via the exponential map and orthographic projection) and matching univariate CDFs to update sample locations iteratively in the Radon domain. Key contributions include a non-gradient, flexible procedure that combines composite trapezoidal CDF approximation with inverse-CDF sampling and multi-projection updates to produce high-quality deterministic samples for any density on the circle, scalable beyond conventional UKF-like sample counts. The approach promises improved state estimation with fewer samples and lays groundwork for extensions to higher-dimensional manifolds such as the hypersphere and torus, using adaptive numerical integration and projection-aware strategies.

Abstract

We propose a method for deterministic sampling of arbitrary continuous angular density functions. With deterministic sampling, good estimation results can typically be achieved with much smaller numbers of samples compared to the commonly used random sampling. While the Unscented Kalman Filter uses deterministic sampling as well, it only takes the absolute minimum number of samples. Our method can draw arbitrary numbers of deterministic samples and therefore improve the quality of state estimation. Conformity between the continuous density function (reference) and the Dirac mixture density, i.e., sample locations (approximation) is established by minimizing the difference of the cumulatives of many univariate projections. In other words, we compare cumulatives of probability densities in the Radon space.

Figures (5)

• Figure 1: Wrapped Laplace Distribution (blue) on the circular domain (black), with proposed deterministic sampling result for 35 samples (red).
• Figure 2: Continuous density function (blue) and two orthographic projections or marginals (yellow, purple), see (\ref{['eq:proj:orth']}).
• Figure 3: Procedure for deterministic sampling of a projected von Mises-Fisher density, using orthographic projection (\ref{['eq:proj:orth']}). Upper part: we evaluate $f(r)$ (blue) at the fixed evaluation points $t_j^{{\operatorfont{h}}}$ (black) as well as previous sample locations (red). Lower part: Trapezoidal integration on said evaluation points is performed (blue). Compare the ground truth obtained with a numerical ODE solver (yellow). Then, one-dimensional deterministic sampling is performed (black), yielding an approximating DM distribution function (red). See also Alg. \ref{['alg:sampleProjected']} for a more detailed description.
• Figure 4: Illustration of various circular distributions and deterministic samples obtained with the proposed method. Continuous probability density function (blue) on the angular domain $S^1$ (black), with sampling results (red). For better visualization, the length of the red lines representing the unweighted samples has been set to the maximum density function value (mode) instead of the sample weight $1/L$.
• Figure 5: Deterministic circular sampling using (a) only a fixed set of 30 evaluation points $t_j^{{\operatorfont{h}}}$ versus (b) the 30 fixed points plus the previous samples, for better numerical integration. The difference for this quite "narrow" von Mises-Fisher distribution ($\kappa=500$) is notable.